This correction has been studied in Refs. [1]} and [2]} to relative order \(1/N_c^3\)  in the \(1/N_c\)  expansion for \(\Delta =0\)  and \(\Delta \ne 0\) , respectively. For definiteness, in Eq. (REF ), the explicit sum over spin \(\mathsf {j}\)  is indicated but the sums over spin and flavor indices are understood, the baryon axial current operators \(A^{ia}\)  and \(A^{jb}\) , Eq. (REF ), are used at the meson-baryon vertices, \(\mathcal {P}_{\mathsf {j}}\)  is a spin projection operator for spin \(J=\mathsf {j}\) , and \(\Gamma ^{ab}(\Delta _{\textsf {j}})\)  is an antisymmetric tensor which depends on the difference of the hyperfine mass splitting for spin \(J=\mathsf {j}\)  and the external baryon. The most general form of \(\mathcal {P}_{\mathsf {j}}\)  for arbitrary \(N_c\)  can be found in Ref. [3]}. The spin-\(\frac{1}{2}\)  and spin-\(\frac{3}{2}\)  projectors for \(N_c=3\)  required here reduce to
\(\mathcal {P}_\frac{1}{2} & = & -\frac{1}{3} \left[ J^2 - \frac{15}{4} \right], \\\mathcal {P}_\frac{3}{2} & = & \frac{1}{3} \left[ J^2 - \frac{3}{4} \right],\) 
